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I am given a situation where $X_1,X_2,...X_n \sim_{iid} X$ with pdf $f_X(x;\theta)$ and $u(X)$ is an unbiased and sufficient estimator of $\theta$.

I am working on proofs where I am trying to find if $X$ is complete and it seems that as long as

$E[u(X)]=0$ is given and you can show that $E[X^ku(X)]=0$ for any natural number $k$ then you can show $u(X)=0$.

I do not know why the last logic works.

May I have some help please?

hyg17
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  • What kind of integrability assumptions do you have on $X$ and $u$? – saz Jul 01 '19 at 06:43
  • I was not given any information that would answer your question, but I am not working with any "special situation" where we are working with unusual distributions. So far, I have seen this same explanation with exponential distribution, normal distribution and Poisson distribution so I cannot imagine there being any issues with integrability. – hyg17 Jul 01 '19 at 16:45
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    Here and here are related questions. By assumption, $E(p(X)u(X))=0$ for any polynomial $p$ and under suitable assumptions this gives $E(g(X)u(X))=0$ for any continuous function $g$. This in turn gives $E(1_B(X) u(X))=0$ for any measurable set $B$; now consider $B={u>0}$ and $B={u<0}$. – saz Jul 01 '19 at 17:50
  • I see. It looks like it is in the field of real analysis. Unfortunately it is beyond my understanding and I cannot proceed from here, but it does seem to be a fact that is well supported and used. For now I will accept this as a fact that I have to use blindly. – hyg17 Jul 01 '19 at 18:02

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